Let $\alpha$ be an irrational number. The task is to prove that there exist infinitely many rational numbers $\frac{p}{q}$ (where $p$ and $q$ are integers) satisfying the following inequality:

$\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$

This result is a famous theorem from Diophantine approximation known as Dirichlet's approximation theorem. The theorem guarantees that for any irrational number $\alpha$, there are infinitely many rational numbers $\frac{p}{q}$ such that:

$\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$

Proof (Outline of Dirichlet's Approximation Theorem):

1. The pigeonhole principle:
   Consider the fractional parts $\{ n\alpha \}$ for $n = 1, 2, \dots, Q$, where $\{ x \}$ denotes the fractional part of $x$. These are values in the interval $[0, 1)$, and there are $Q$ such values.

2. Partition the interval:
   Divide the interval $[0, 1)$ into $Q$ equal subintervals of length $\frac{1}{Q}$. By the pigeonhole principle, at least two of the fractional parts $\{ n_1 \alpha \}$ and ( { n_2 \alpha }